Through the previous steps, the source domain distribution is \(P_s = \sum _{i=1}^{k_s} w_{s,i} p_{s,i}\)  while the target domain distribution is \(P_t = \sum _{i=1}^{k_t}w_{t,i} p_{t,i}\) . And the label corresponding to \(p_{s,i}\)  is the label of the data belongs to \(i\) th cluster in the source domain, i.e. \(\tilde{y}_{s,i}\) . According to Equation equation REF  or Equation equation REF , we can easily get the cost matrix \(\mathbf {C}\) .
Following [1]}, the objective for SOT is
\(\begin{aligned}\pi ^* &= &\arg \min _{\pi } \langle \pi , \mathbf {C}\rangle _F + \lambda H(\pi ) + \eta \Omega (\pi ) \\s.t& &\pi ^T \mathbf {1}_{k_s} = \mathbf {w}_t\\&&\pi  \mathbf {1}_{k_t} = \mathbf {w}_s.\end{aligned}\) 
